s₀=0.0, s_i+1=-0.5*s_i²+2.8026
Center:-0.33725+0.00007i; Zoom = 5120

Another drop cloth fractal. The same generating formal as yesterday, but with a different seed. I have made minor tweaks to the location, zoom and color.

s₀=0.4, s_i+1=-0.5*s_i²+2.8026
Center:-0.33742+0.00008i; Zoom = 2560

Now a very small change to the adder in the sequence generating formula. The capture set is gone. The big picture, zoom = 1, look very much like Sequence Fractals Part IV #18 except the main body is no longer white. It is a palette-dependent non-capturing color.

By my spreadsheet investigation, The sequence converges slowly to a 32-cycle. It may have bifurcated to a 64-cycle.

The result is on the boundary between normal swirly fractals and the jigsaw fractals. It reminds me of a paint drop cloth after a sloppy painter has used it for many projects.

s₀=0.2.36, s_i+1=-0.5*s_i²+2.8
Center:-0.359+0i; Zoom = 320

Same generating sequence as the previous few pictures. With the starting point selected to speed up the convergence of the cycle. (The starting point is within 0.01 of one of the points on the 32-cycle.) As expected, the quicker convergence puts the sequence closer to a simple repeating (although somewhat long cycle) sequence, and the capture set is larger.

s₀=0.4, s_i+1=-0.5*s_i²+2.8
Center:0.0305+0.2902i; Zoom = 410

Zoom of Sequence Fractals Part IV #18

s₀=0.4, s_i+1=-0.5*s_i²+2.8
Center:0.0311229+0.2895398i; Zoom = 105000

A zoom into yesterday’s picture Sequence Fractals Part IV #18

I had put this formula into a spreadsheet and generated the first forty values. There were no patterns, It appeared to be a chaotic sequence. In other situations a chaotic sequence leads to “jigsaw puzzle fractals”. Long story short, the sequence is slowly converging to a long cycle. When I extended the spread sheet to 300 rows the sequence appears to be a freshly-bifurcated 32 cycle.

If one were to combine the 32 steps like we did with 2-cycle sequences, we would get a billion degree polynomial. As ridiculous as that number sounds, the pictures do seem normal, about what you would expect from a polynomial.

s₀=0.4, s_i+1=-0.5*s_i²+2.8
Center:-0+0i; Zoom = 1

Now the sequence is generated by a*s_i²+b, a and b real numbers. You may be familiar with sequences like this, the Logistic Map and the real axis of the Mandelbrot set are two examples.

The parameters can be tweaked to produce a variety of behaviors in the sequence. For a < 0 and b > 0 and small the sequence will converges to a point. As b increases it goes through a period doubling phase where is converges to a 2 cycle, then 4, 8 as so on. Eventually after hitting every power of two, it becomes chaotic. The Wikipedia page linked above describes this phenomena in more detail.

s₀=1.0, s_i+1=(-0.96+0.28i)*s_i
Center:0.138+0.0.402i; Zoom = 64

A small zoom into the previous picture, Sequence Fractals Part IV #16.

s₀=1.0, s_i+1=(-0.96+0.28i)*s_i
Center:0.121+0.419i; Zoom = 32

Here is a zoom into the top edge of the circle in the previous picture, Sequence Fractals Part IV #15.

I call these jigsaw puzzle fractals. They does not have the normal behavior where just a few points around the edge of the capture set escape each step, resulting in smooth colors and swirls. Instead large chunks escape on distinct iteration steps. The result looks like fitting together distinctly colored puzzle pieces.

Here are some earlier examples of “jigsaw puzzle” fractals: Sequence Fractals Part III #27, Sequence Fractals Part III #38

s₀=1.0, s_i+1=(-0.96+0.28i)*s_i
Center: 0+0i; Zoom = 0.5

Next up: Non-periodic sequences. This one is based on the 7-24-25 Pythagorean triple. The Pythagorean triples provide an easy way to find rational coordinates that generate an irrational angle. The sequence moves around the circle with radius 1.0 by a rotation of an irrational angle.

I used a similar sequence with earlier, Sequence Fractals Part III #27 for example.

With the ‘*s_i‘ formula, almost all of these look like a circle. That makes some sense. The sequence s_i is dense in the unit circle. Informally that means it is everywhere. Formally, pick any point on the circle and your favorite small and there is a point in the sequence within of your point.

All c values (pixels) with a common radius will behave similarly. |*s_i| = 1.0. If r > 0, |c| = r, then the adder at each step, +c*s_i has |c*s_i| = r. The c*s_i are dense in the circle with radius r. For the most part, only |c| matters; all c’s on a given circle have the same eventual fate, capture or escape. It is only near this circular capture/escape border the angle of c makes a difference.

If the colors were adjusted they would look like the sun with solar flares.

s_i = 1, -.5, 1 (repeating)
Center:0.58873-0.26113i , Zoom = 4096

Zoom into the left side of yesterday’s image, Sequence Fractals Part IV #13.